OFFSET
0,4
FORMULA
a(n) = det(L_n+U_n-I_{n+1}), where L_n is the lower triangular Pascal matrix of order n, U_n is the transpose of L_n and I_n is the identity matrix of order n. Note that L_n, U_n and I_n all have determinant 1 for all n.
EXAMPLE
a(0) is the determinant of the 1 X 1 matrix whose sole entry is one.
a(1) is the determinant of the 2 X 2 matrix of all ones.
a(2) is the determinant of the 3 X 3 matrix
[1 1 1]
[1 1 2]
[1 2 1].
a(3) is the determinant of the 4 X 4 matrix
[1 1 1 1]
[1 1 2 3]
[1 2 1 3]
[1 3 3 1].
MATHEMATICA
PascalMatrix = Function[n, Table[Table[Binomial[m, i], {i, 0, n}], {m, 0, n}]];
PascalDet = Function[n, Det[PascalMatrix[n] + Transpose[PascalMatrix[n]] - IdentityMatrix[n + 1]]];
Table[PascalDet[i], {i, 0, 19}]
PROG
(Python)
from sympy import *
def m(N):
return Matrix([
([binomial(i, n) for n in range(i+1)] +[0] * (N-i))
for i in range(N+1)
])
def matrix(N):
return m(N) + m(N).transpose() - eye(N+1)
[ matrix(i).det() for i in range(20)]
# Gilles Castel, Sep 25 2017
CROSSREFS
KEYWORD
sign
AUTHOR
Alexander Farrugia, Sep 25 2017
STATUS
approved